sum of square-free divisors

[Pieter Moree]

Steven Finch wrote:

> Given a positive integer n, let a(n) denote the sum of
> the
>  square-free divisors of n, including 1.  For example,
>
> a(1000) = 18 = 1+2+5+10.
>
> The sequence {a(n)} is A048250 in Sloane's database
>
> http://www.research.att.com/~njas/sequences/
>


> Someone has suggested to me that sum_{n<=x) a(n) should be of the form
>  C*x^2. Is this true?  If yes, what exactly is the constant C?
C=1/2.

The proposition is

\sum_{n\le x}a(n)=x^2/2+O(x\sqrt{x}).
I checked it on maple. Seemed consistent.

As S. Finch pointed out to me, in
D. Suryanarayana, On the core of an integer, Indian J. Math. 14 (1972)
65--74

this is proved under the Riemann hypothesis with error term
O(x^{7/5+\epsilon}).

Pieter Moree